UNRAVELING THE INTEGRAL KNOT CONCORDANCE GROUP

This paper gives a detailed description of the design and structure

of the integral knot concordance group and solves the realization prob-

lem for the algebraic invariants described by J. Levine in his fundamen-

tal paper, "Invariants of Knot Cobordism", [L2] in which he demonstrates

their sufficiency. The geometric problem underlying and motivating the

algebraic discussion of this paper is the classification of spherical

knots under the equivalence relation of concordance: A smooth, oriented

manifold pair (M ,K ) , where K is a codimension two embedded sub-

manifold of M , is a spherical knot if M and K are homotopy spheres.

Two such pairs with the same ambient manifold M are concordant if

there is a proper oriented codimension two h-cobordism H in M x

[0,1] (with dH c d(MXI) such that the restriction to M X {0,1} is

the pairs (M,Kn) and (M,-K-, )) (where -denotes the opposite orienta-

tion). This is the geometric knot concordance group C- (s) first

introduced by Fox and Milnor [FM] for the case n = 1 . Kervaire com-

puted that the group was trivial for n even in [Kl] and Levine defined

an algebraic obstruction group (which we denote C (^) after Kervaire

[K2]) which he demonstrated was isomorphic to the geometric group

^2 +1^ for e = (""^ provided n 1 in [Ll]. (For n = 1 there

is only a surjection from the geometric group.) In [L2], Levine computed

the analogous group, C (Q) over the field of rational numbers (with

the aid of a computation of Milnor [Ml]) and observed that the integral

group injected into the rational group.

The group arises in a concrete geometric manner from the classical

Seifert pairing. If (M,K) is a spherical knot, then K is the

boundary of a framed codimension one submanifold V of M . The inter-

section pairing on the middle dimensional homology of V is e =

(-1)k

symmetric if n = 2k - 1 and is unimodular (has a unit determinant)

Received by the editor November 1, 1976.

This research was partially supported by an NSF Grant.